Cyclic flats of binary matroids

Freij-Hollanti, Ragnar; Grezet, Matthias; Hollanti, Camilla; Westerbäck, Thomas

doi:10.1016/j.aam.2021.102165

Cited by 6 publications

(6 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where the last equality follows from the assumption g ∈ C(U ). Since C is non-degenerate, from (12) we have that ag r = 0, we get a contradiction. This shows that the C(V ) = C(U ) for every subspace U of V of codimension 1 and, in other word, the support of every codeword in dual code of C is a cyclic space in the matroid M .…”

Section: Rank-metric Codes and Q-matroidsmentioning

confidence: 94%

“…Brylowski outlined in [4, Proposition 2.1] an algorithm for constructing the lattice of flats of a matroid from its lattice of cyclic flats along with their ranks. In [12,Section 5], the authors also showed how to reconstruct the lattice of flats from the lattice of cyclic flats, along with their ranks. The same construction applies in the q-analogue.…”

Section: Cyc(cl(a)) + a = Cl(a)mentioning

confidence: 99%

“…where, the intersection of an empty set of spaces is equal to the whole space E. Clearly, if F is a flat then F ∨ = F . Then, as in [12,Proposition 6], if F ∨ ≤ F , we have that F is a flat if and only if for every cyclic flat A satisfying…”

Section: Cyc(cl(a)) + a = Cl(a)mentioning

confidence: 99%

“…In fact, it has been proved that many central invariants in coding theory can be naturally described in terms of the lattice of cyclic flats of the associated matroid. This was shown in relation to distributed data storage; see [12,26]. Furthermore, even the lattice of cyclic flats of classical polymatroids has been considered; see [10,27,28].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Cyclic Flats of a $q$-Matroid

Alfarano¹,

Byrne²

2022

Preprint

View full text Add to dashboard Cite

In this paper we develop the theory of cyclic flats of q-matroids. We show that the lattice of cyclic flats, together with their ranks, uniquely determines a q-matroid and hence derive a new q-cryptomorphism. We introduce the notion of F q m -independence of an F qsubspace of F n q and we show that q-matroids generalize this concept, in the same way that matroids generalize the notion of linear independence of vectors over a given field.

show abstract

Section: Rank-metric Codes and Q-matroidsmentioning

confidence: 94%

Section: Cyc(cl(a)) + a = Cl(a)mentioning

confidence: 99%

Section: Cyc(cl(a)) + a = Cl(a)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Cyclic Flats of a $q$-Matroid

Alfarano¹,

Byrne²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…In [13, Section 5], the authors also showed how to reconstruct the lattice of flats from the lattice of cyclic flats, along with their ranks. The same construction given in [13] applies in the q-analogue and we give a brief sketch. For each X ∈ L(E), define two cyclic flats…”

Section: Cyclic Flatsmentioning

confidence: 99%

The cyclic flats of a q-matroid

Alfarano,

Byrne

2024

J Algebr Comb

View full text Add to dashboard Cite

In this paper we develop the theory of cyclic flats of q-matroids. We show that the cyclic flats, together with their ranks, uniquely determine a q-matroid and hence derive a new q-cryptomorphism. We introduce the notion of $$\mathbb {F}_{q^m}$$ F q m -independence of an $$\mathbb {F}_q$$ F q -subspace of $$\mathbb {F}_q^n$$ F q n and we show that q-matroids generalize this concept, in the same way that matroids generalize the notion of linear independence of vectors over a given field.

show abstract